Ultrasonic measurement and methods for reconstruction of temperature fields for the use in bioreactors

3.1 Setup for measuring temperature fields

In order to be able to measure the average temperature along a sufficiently large number of measurement paths in one plane, several transducers are required, which are positioned at the boundary of the plane under consideration. For the number of transducers, a compromise must be made between the number of measurement paths and the increasing effort. In the following, a setup is considered in which 16 transducers are attached at regular intervals along the circumference of a round, water-filled basin. This results in a total of 120 theoretically usable measurement paths. However, since the used transducers of the type SMFM21F1000 from Steiner & Martins with a resonant frequency of 1 MHz have highly directional emission characteristics [7], [8], not all theoretical paths can be used in reality. Figure 3 (a) shows the remaining usable measurement paths for one transmitter. Consequently, there are 56 usable paths for all 16 transducers. In order to be able to use more information, the paths shown as solid lines in Fig. 3 (b) with backscattering on the opposite basin wall are additionally used. This results in 32 additional paths for all transducers and thus in a total of 88. The paths shown as dotted lines provide no additional information since the location of the backscattering is on the surface of another transducer. Therefore, these paths are not considered.

Figure 3

Direct measurement paths (a) that can actually be used, shown for one transmitter, and additionally usable paths with backscattering (b) for a circular arrangement of 16 transducers. The dotted lines in (b) show paths that provide no additional information and are therefore not used.

A photo of the basin with 16 glued-in transducers is shown in Fig. 4. The basin is made of acrylic glass and has an inner diameter of 194 mm and a height of 208 mm. The 16 transducers, which are mounted in one slice plane of the basin, must all be able to be used as both transmitters and receivers. For this purpose, all transducers are connected to electronics for switching between transmission and receiving mode. This electronics consists of 16 relays and a 16:1 multiplexer, which are all controlled by a microcontroller. The relays are used to switch between transmission and receiving mode. The multiplexer is used to select the receiver to be connected to an oscilloscope that is used to record the received signals. Although these electronics only allow sequential measurements to be carried out and are not designed for the shortest possible measurement time, this is sufficient under laboratory conditions. However, the measurement time can be significantly reduced by acquiring the received signals from several transducers in parallel. With suitable hardware and depending on the dimensions of the measurement area, measurement times of a few seconds or even less than 1 s can be achieved. The transmission signals are generated using an STHV748 ultrasonic pulser manufactured by STMicroelectronics. The transmission signal has a frequency of 1 MHz and consists of 10 rectangular pulses followed by 5 pulses with a lower magnitude that are delayed by half a period [9], [10]. Due to the number of pulses, the ultrasonic wave package has enough energy to ensure reliable detection. The time-delayed pulses lead to a faster decay of the transducer’s oscillation.

Figure 4

Basin made of acrylic glass with an inner diameter of 194 mm and 16 glued-in transducers with a resonant frequency of 1 MHz distributed evenly along the basin’s circumference for measuring temperature fields in the slice plane of the transducers [11]. Above the transducers a Pt1000 temperature sensor can be seen, which is used for reference measurements.

The next section deals with the further processing of the received signals to determine the travel times.

3.2 Determination of travel times

The received signals from all 88 measurement paths are processed with an FIR bandpass filter. Figure 5 shows the filtered received signals of a direct measurement path at two different temperatures. The shorter travel time at the higher temperature is clearly visible.

Figure 5

Filtered received signals of a single measurement path at two different temperatures.

To determine the ultrasonic travel times for all 88 measurement paths, the filtered signals are compared with reference signals of known travel times. For this purpose, the cross-correlation of the received signal and the reference signal is calculated for each path. With the shift against 0 of the maximum of the resulting cross-correlation function, the delay of the signal with respect to the reference signal can be estimated [12]. Since the noise that remains in the relevant frequency range around 1 MHz after filtering the signals is approximately white, cross-correlation is a matched filter. Therefore, the use of cross-correlation to determine the travel times yields very good results. The standard deviation of the travel time measurement is less than 1.2 ns for all paths. With the measurement setup described above, this corresponds to a standard deviation of the temperature measurement of less than 5 mK at room temperature.

For the reconstruction dealt with in the next section, the average speed of sound values along all measurement paths are calculated from the travel times determined with cross-correlation according to Eq. 6.

4.1 Discretization of reconstruction area and formulation as inverse problem

The circular reconstruction area is subdivided into cells of equal area using a method from [13]. Thereby, the reconstruction area is divided into a small inner circle and N rings. These rings are then again divided into individual cells. This results in a total of ( 2 N + 1 ) 2 cells. This discretization is shown in Fig. 6 for N = 3 rings as an example, resulting in 49 cells.

Figure 6

Example of the discretization of a circular area in cells of equal area for N = 3 rings [6]. The cells are denoted with x 1 … x 49 .

The inverse problem to be solved for the reconstruction of the distribution of the speed of sound, from which the temperature field is then determined, is

The solution vector x contains the values sought for the individual cells of the discretization. Its elements are assigned to the cells according to the designations in Fig. 6. The measurement vector b contains the average speed of sound values of all measurement paths weighted with the respective path length. The system matrix A connects the two vectors x and b. Its elements each contain the length of the measuring paths per cell of the discretization. The measurement matrix A is thus given by the geometry of the measurement setup.

To solve the inverse problem in Eq. 7, the two methods described in the next two sections are used. The temperature field is then calculated from the reconstructed distribution of the speed of sound. Since the 5th order polynomial in Eq. 2 cannot be solved unequivocally for the temperature, this is done iteratively.

4.2 Tikhonov regularization

The difficulty in solving inverse problems is that they are usually ill-posed and therefore even small perturbations in the measurement vector b lead to large errors in the solution x [14]. Since b contains noisy measurement data, these perturbations are always present. Therefore, the naive solution of the inverse problem usually yields very poor results. The Tikhonov regularization tries to counteract this. The Tikhonov solution x λ is the solution of

The first term Ax − b 2 2 in Eq. 8 measures the goodness-of-fit of the solution. The second term x 2 2 measures the regularity of the solution. Since many direct measurement problems exhibit low-pass behavior, high-frequency components are amplified more compared to lower-frequency components in the associated inverse problems [14]. Consequently, the naive solution of Eq. 7 is dominated by high-frequency noise in b. By the term x 2 2 , this is taken into account in the calculation of the Tikhonov solution according to Eq. 8. The parameter λ weights the two terms in Eq. 8 with respect to each other [15]. For λ = 0, the naive solution results from Eq. 8. The quality of the reconstruction highly depends on the choice of the parameter λ, which is why the best possible value for it has to be found for good reconstruction results.

4.3 Compressed sensing

Methods known as compressed sensing [16] can also be used to solve inverse problems given certain a priori knowledge. The condition for using these methods is that the solution vector x is sparse or at least compressible or can be transformed into a respective representation [17], [18], [6]. A vector x is called sparse if x i ≠ 0 holds for only a few of its elements. Sparsity is rarely given in real applications, e. g. due to noise. Instead, only a few elements are significantly different from zero and the rest have values close to zero. A vector for which this is the case is called compressible. Compressible vectors minimize the ℓ 1 norm

Minimizing the ℓ 1 norm also often provides the sparsest solution, according to [19].

A temperature field is typically not sparse or compressible, and neither is the corresponding speed of sound distribution. However, it can be transformed into a compressible representation if it can be assumed that the distribution has only a few hotspots and is otherwise approximately constant. This is often the case in bioreactors. The transformation into a compressible representation is then possible by using the gradient of the solution vector. The ℓ 1 norm of the gradient is referred to as total variation and is denoted by ∙ TV [20]. The reconstruction of the distribution of the speed of sound, from which the reconstructed temperature field is calculated, is then done by solving the optimization problem

with the remaining error ϵ. It specifies the termination condition of the optimization [21].

In contrast to Tikhonov regularization, minimization of the total variation does not have a parameter comparable to λ on which the quality of the solution depends significantly. Instead, a priori knowledge is required for total variation minimization so that the gradient of the solution vector can be used for the reconstruction.

5.1 Results from simulated data

To simplify testing and validation of the reconstruction, the travel time measurement is simulated for assumed artificial temperature fields. For this purpose, the distribution of the speed of sound is derived from the temperature field and the travel times are calculated for all measurement paths according to Eq. 5 by numerical integration using Gaussian quadrature. For a more realistic reconstruction, the simulated travel times are overlaid with noise of the order of magnitude of the travel time measurement in Section 3.2.

Figure 7

Reconstruction of a temperature field from simulated ultrasonic travel times using Tikhonov regularization (a) and minimization of the total variation (b) for an assumed distribution with the constant value ϑ = 20 ° C.

Figure 7 shows the reconstruction results of a constant temperature distribution with ϑ = 20 ° C. The result obtained by minimizing the total variation in (b) deviates minimally from 20 °C only because of the noise added to the travel times. This is to be expected, since a constant distribution is maximally compressible. The result with Tikhonov regularization in (a), on the other hand, shows a much larger scattering of the reconstructed values of the cells of discretization. The average of the reconstructed values of the cells is 19.914 °C, the standard deviation is 244 mK. It can be clearly seen that the values of the individual cells are not random, but follow a symmetrical pattern that depends on the positions of the 16 transducers. Therefore, these are artifacts caused by the geometry of the measurement setup and the reconstruction. The resulting systematic error is thus also to be expected when reconstructing other distributions with Tikhonov regularization. Nevertheless, the temperature field with a constant value is in principle still well reconstructed with Tikhonov regularization.

Another assumed temperature distribution is shown in Fig. 8 (a). The distribution consists of a hotter inner ring with temperature ϑ = 21 ° C and a colder outer ring with ϑ = 19 ° C. The remaining area of the distribution has a temperature of 20 °C. This distribution does not represent a case that is to be expected in reality. Due to the rotational symmetry of the distribution, however, this represents a case that is as difficult as possible for the reconstruction from the travel times along the measurement paths, which are also rotationally symmetrical. In addition, the effects of the two rings on the travel times, especially for the paths between two opposing transducers, almost cancel each other out. This makes it even more difficult to reconstruct the rings correctly.

Figure 8

Assumed temperature distribution (a) with adjacent, ring-shaped hot and cold spots near the center. The hotter inner ring has a temperature of ϑ = 21 ° C, the colder outer ring a temperature of ϑ = 19 ° C and the areas inside and outside the rings a temperature of ϑ = 20 ° C. In (b) and (c) reconstruction results are shown with Tikhonov regularization and minimization of the total variation, respectively.

In Fig. 8 (b) the result with Tikhonov regularization can be seen, the result with minimizing the total variation in (c). The reconstruction result obtained by minimization of the total variation is again very good both qualitatively and quantitatively. Both rings are also reconstructed very well with Tikhonov regularization. However, the rest of the reconstructed distribution shows the artifacts seen in the previous example. Moreover, the outer area (outer two rings of the discretization) has a slightly too low value.

This example shows that rotationally symmetrical distributions can be reconstructed using both methods. However, if the radii of the rings are increased, i. e. if they are positioned closer to the outer edge, the reconstruction no longer works. This is due to the fact that data from more different measurement paths are available for the reconstruction near the midpoint. This is also the reason why the outer area has a slightly incorrect value with Tikhonov regularization.

After the reconstruction results from simulated data above, a result from measured travel times follows in the next section.

5.2 Results from measured data

For the reconstruction of a temperature field from real data, the travel times are measured using the setup from Section 3.1. The distribution to be measured has a hotspot that is generated by locally introducing heat. For this purpose, a coiled resistance wire made of constantan is used, to which current pulses are applied. The center of the spot where the heat is introduced has approximately the coordinates x = 2 mm and y = − 38 mm.

Figure 9

Reconstruction of a temperature field with a hotspot from measured ultrasonic travel times using Tikhonov regularization (a) and minimization of the total variation (b) [6], [11]. The hotspot was created by locally introducing heat in a small area around x = 2 mm and y = − 38 mm.

The reconstruction results achieved with Tikhonov regularization and minimizing the total variation are shown in Fig. 9. The position of the generated hotspot is reconstructed correctly and unambiguously with both methods. The result obtained by minimizing the total variation in (b) again shows a significantly lower scattering of the reconstructed distribution outside the hotspot and the hotspot can be recognized much more clearly. With Tikhonov regularization, the values in the outer area are again somewhat too low, as already seen in the example in Fig. 8. However, Tikhonov regularization is also well suited for localizing the hotspot.

5.3 Measurement uncertainties

As already discussed in Section 3.2, the standard deviation of the travel time measurement is less than 1.2 ns for all paths, which corresponds to a standard deviation of the temperature measurement of less than 5 mK, assuming the size and shape of the investigated vessel. Additional deviations result from the non-linearity of the relationship between speed of sound and temperature.

The results from simulated data in Section 5.1 showed, that for small hotspots with small temperature deviations from the mean, a reconstruction of temperature fields for water with an uncertainty of about 0.1 K is possible with minimization of the total variation and with an uncertainty of about 0.5 K with Tikhonov regularization. Further errors can arise from the extent of hotspots not being congruent to the discretization. However, since these errors depend on the distribution to be reconstructed, they cannot be generally quantified. In the case of measurements in real bioreactors, the reactions forming products from reactants with their inherently different speeds of sound can also cause small additional uncertainties.